What is the Sampling Period Range for Recoverable Signals?

[satchmo05]

Homework Statement

The signal y(t) is generated by convolving a band limited signal x₁(t) with another band limited signal x₂(t) that is y(t)=x₁(t)*x₂(t) where:

--> X₁(jω)=0 for|ω| > 1000Π
--> X₂(jω)=0 for|ω| >2000Π

Impulse train sampling is performed on y(t) to obtain:
--> y_p(t)= [summation from n = (−∞,∞)] y(nT)δ(t− nT)

Specify the range of values for sampling period T which ensures that y(t) is recoverable from y_p(t).

Homework Equations

All of the equations that I would are most likely showing.

The Attempt at a Solution

My thoughts were to plug in (nT) for every t in both x₁(t) and x₂(t) and then take the Fourier transform of that, cut of the edges where the transforms are equal to zero and then that is where I go blank...

I imagine that is the right implementation to start the problem with, but please correct me if I am wrong. Thank you in advance to all who may be able to help - it is much appreciated!

 

[vela]

Hint: convolution theorem.

 

[satchmo05]

I can see where convolution comes into play, but how can I implement the CTF transforms that are given?

 

[vela]

What sets the lower limit on the sampling rate if you want to be able to recover the original signal?

 

[satchmo05]

The Nyquist rate, to sample at the perfect rate (without aliasing/oversampling to occur) - it would be = 2fmax

 

[satchmo05]

I am pretty confused at what you're trying to hint at here. I appreciate the help, but my mind is still blank.

 

[vela]

It's kind of hard to say anything without giving away the answer. Think about Y(jω). Where is it zero? Can you deduce f_max from that information?